I went for the middle ground and had a bass made by Adrian Maruszczyk via his custom shop configurator. Choices are not infinite but more than enough to get a bass that suited me - a lightweight 5 string Precision with a narrow neck. Fender didn't make one so this was perfect. At every stage you get plenty of options for material and finish for body, neck, fingerboard, neck markers, binding, headstock, frets, hardware, electrics, bridge. So not exactly an actual custom made bass as it is made within certain parameters, but you end up with a bass that is pretty exactly as you specify for a good deal less than an actual custom shop job. Mine arrived on my doorstep after a 3 month wait and cost a whisker under £1500 including delivery, a SKB hard case and leather strap.
I couldn't be happier with it, I must say. But then I quite like my basses to be traditional shapes. If I wanted one of those cartoon whale single cuts or something that looks like a piece from a jig-saw puzzle this route wouldn't suit.